Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,12)(13,21)(14,15)(16,17)(18,22)(19,23)(20,24)(25,26,31,28)(27,32,34,30)(29,36,38,33)(35,37,40,39);(3,7)(6,8)(10,15)(11,17)(18,21)(19,24)(25,27,31,34)(26,32,28,30)(29,35,38,40)(33,39,36,37);(1,2,5,4)(3,6,7,8)(10,15)(11,17)(18,21)(19,24)(26,28)(27,34)(33,36)(35,40);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28,31,26)(27,32,34,30)(29,33,38,36)(35,37,40,39);(1,4,5,2)(3,8,7,6)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,38)(26,33,28,36)(27,35,34,40)(30,37,32,39);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30,31,32)(26,34,28,27)(29,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081543);